How do you convert 0.12345 (345 repeating) to a fraction?

1 Answer
George C.
Mar 28, 2016

#0.12bar(345) = 4111/33300#

Explanation:

Multiply by #100(1000-1) = 100000-100# to get an integer:

#(100000-100)0.12bar(345)=12345.bar(345)-12.bar(345)=12333#

The factor #100# shifts the number so that the repeating portion starts just after the decimal point. The factor #(1000-1)# shifts it left by a further #3# places (the length of the repeating pattern) and subtracts the original to cancel the fractional tail.

Then divide both ends by #(100000-100)# to get:

#0.12bar(345) = 12333/(100000-100) = 12333/99900 = (color(red)(cancel(color(black)(3)))*4111)/(color(red)(cancel(color(black)(3)))*33300) = 4111/33300#

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